Nonlinear elastic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise because nonlinear elastic materials undergo large reversible deformations, and often possess complex micro-structures. In this work, we propose and explore an alternative approach to model finite elasticity problems in two dimensions by using polygonal discretizations. To account for incompressible behavior, a general theoretical framework of deriving the two-field variational principle, involving a displacement field and a pressure field, is presented. Within the theoretical framework, by assuming different forms of stored-energy function, two types of variational principles are obtained, each with distinct definitions of the independent pressure field. Based on the theoretical setting, we present both lower order displacement-based and mixed polygonal finite element approximations, the latter of which consist of a piecewise constant pressure field and a linearly-complete displacement field at the element level. Through numerical studies, the mixed polygonal finite elements are shown to be stable and convergent. Finally, in the context of filled elastomers and cavitation instabilities, we present applications of practical interest, which utilize polygonal discretizations, demonstrating the potential of polygonal finite elements in studying and modeling nonlinear elastic materials of complex micro-structures under finite deformations.
PDF
download
878987797
028-85220240
